[PPT] - Communication network for the GPS III system Simon Crevals Prof. PowerPoint Presentation

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Communication network for the GPS III system

Simon Crevals

Prof. dr. G. Brinkmann
N. Van Cleemput

Department of Applied Mathematics and Computer Science Ghent University

July 2010

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1

Introduction

2

First test

3

Final program

4

Results

5

Conclusion

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1

Introduction

2

First test

3

Final program

4

Results

5

Conclusion

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Advantages of communication

continuous telemetry frequent updates

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Restrictions for connections

connection possible during the entire orbit 4 connections a satellite

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Definition

VGPS is the set of all satellites.

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Definition

VGPS is the set of all satellites.

Definition

GPos = (VGPS, EPos), with ∀v, w ∈ VGPS : {v, w} ∈ EPos ⇔ communication between satellites v and w is technically possible.

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Definition

VGPS is the set of all satellites.

Definition

GPos = (VGPS, EPos), with ∀v, w ∈ VGPS : {v, w} ∈ EPos ⇔ communication between satellites v and w is technically possible.

Definition

A connection graph is a spanning, 4-regular subgraph of GPos.

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Definition

Given a graph G = (V , E) and a vertex v ∈ V : the neighbourhood of v in G is N(v, G) = {w ∈ V |{v, w} ∈ E}.

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Definition

Given a graph G = (V , E) and a vertex v ∈ V : the neighbourhood of v in G is N(v, G) = {w ∈ V |{v, w} ∈ E}.

Definition

The function uredop of a vertex v with respect to its neighbours is defined as: Let M4 = {M ⊆ VGPS||M| = 4}. uredop : VGPS × M4 → R

+ {∞}, so that

∀v ∈ VGPS : M ⊂ N(v, GPos) ⇒ uredop(v, M) = ∞.

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Definition

The uredop value of a vertex v in a connection graph G is defined as: uredop(v, G) = uredop(v, N(v, G)).

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Definition

The uredop value of a vertex v in a connection graph G is defined as: uredop(v, G) = uredop(v, N(v, G)).

Definition

The uredop value of a connection graph G is defined as: uredop(G) = max{uredop(v, G)|v ∈ VGPS}.

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Minimum requirements

Construct a connection graph G which is 4-regular is a subgraph of GPos has diameter at most 4 has uredop(G) < 3

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Best connection graph G

Diameter 3 Smallest possible value for uredop(G) Maximum diameter 4 after one edge removal

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t

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t

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Questions?

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1

Introduction

2

First test

3

Final program

4

Results

5

Conclusion

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Questions

Are there connection graphs with diameter 3? How many? With which uredop value?

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Method

Generate 4-regular graphs Filter graphs with diameter 3 Determine whether they are subgraph of GPos Determine uredop value

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Results

Thousands of millions of 4-regular graphs with 27 vertices and diameter 3 Almost all tested graphs were subgraph of GPos Very different uredop values (also good ones)

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Results

Thousands of millions of 4-regular graphs with 27 vertices and diameter 3 Almost all tested graphs were subgraph of GPos Very different uredop values (also good ones)

Conclusion

The uredop values will have to provide the biggest restriction.

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1

Introduction

2

First test

3

Final program

4

Results

5

Conclusion

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Definition

Given a graph G = (V , E). A vertex set IS ⊆ V is an independent set in G ⇔ ∀v, w ∈ IS : {v, w} / ∈ E.

Definition

Given a graph G = (V , E). A vertex set C ⊆ V is a clique in G ⇔ ∀v, w ∈ C : {v, w} ∈ E. If there doesn’t exist a clique C ′ with |C ′| > |C|, then C is a maximum clique in G.

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Definition

GC(u) = (VC(u), EC(u)), with VC(u) = {(s, N)|s ∈ VGPS, N ⊂ N(s, GPos), |N| = 4 and uredop(s, N) < u} and EC(u) = {{v, w}|v = (s, Ns) ∈ VC(u), w = (t, Nt) ∈ VC(u), s = t and s ∈ Nt ⇔ t ∈ Ns}.

1

13 7 5 2

3

13 9 5 2

2

21 9 7 3

2

12 9 3 1

1

15 7 5 3

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Definition

GC(u) = (VC(u), EC(u)), with VC(u) = {(s, N)|s ∈ VGPS, N ⊂ N(s, GPos), |N| = 4 and uredop(s, N) < u} and EC(u) = {{v, w}|v = (s, Ns) ∈ VC(u), w = (t, Nt) ∈ VC(u), s = t and s ∈ Nt ⇔ t ∈ Ns}.

Theorem

The largest cliques in GC(u) have at most 27 vertices.

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Definition

GC(u) = (VC(u), EC(u)), with VC(u) = {(s, N)|s ∈ VGPS, N ⊂ N(s, GPos), |N| = 4 and uredop(s, N) < u} and EC(u) = {{v, w}|v = (s, Ns) ∈ VC(u), w = (t, Nt) ∈ VC(u), s = t and s ∈ Nt ⇔ t ∈ Ns}.

Theorem

The largest cliques in GC(u) have at most 27 vertices.

Theorem

Every connection graph G = (VGPS, E) with uredop values smaller than u, corresponds to a maximum clique C in GC(u), with |C| = 27.

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Definition

GC(u) = (VC(u), EC(u)), with VC(u) = {(s, N)|s ∈ VGPS, N ⊂ N(s, GPos), |N| = 4 and uredop(s, N) < u} and EC(u) = {{v, w}|v = (s, Ns) ∈ VC(u), w = (t, Nt) ∈ VC(u), s = t and s ∈ Nt ⇔ t ∈ Ns}.

Theorem

Every maximum clique C in GC(u), with |C| = 27, corresponds to a connection graph with uredop smaller than u.

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Definition

The language Uredop is the set of all strings with the structure g ′#c#u, with g ′ represents a graph G ′ c represents all possible sets of 4 neighbours with corresponding value for each vertex u represents a natural number U and for which there exists a subgraph G of G ′, such that C(G) < U.

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Theorem

The language Uredop is NP-complete.

Proof

It is easy to see that Uredop ∈ NP. We still have to prove that each problem in NP can be reduced to Uredop in polynomial time. To prove this, we reduce the language HC (graphs containing a Hamiltonian cycle) to Uredop.

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General clique solver

Too slow for this specific problem

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Pseudocode recursion

Determine central vertex with smallest number of possible neighbourhoods Amount = 0: backtrack For each possible neighbourhood:

◮ choose neighbourhood ◮ adjust lists with possible neighbourhoods ◮ found connection graph or continue in recursion

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Some details

Diameter as bounding criterium Use of bitvectors Reduce iteration length Efficient adjustment of lists with possible neighbourhoods

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Adjust list with possible neighbourhoods

Split lists once every recursion step Only adjust lists of (not yet visited) neighbours GPos Pass the corresponding lists

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Adjust list with possible neighbourhoods

Split lists once every recursion step Only adjust lists of (not yet visited) neighbours GPos Pass the corresponding lists

Split with and without 2

1 2 5 9 13 1 4 7 13 19 1 2 7 19 1 3 5 9 1 2 3 4 13

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Adjust list with possible neighbourhoods

Split lists once every recursion step Only adjust lists of (not yet visited) neighbours GPos Pass the corresponding lists

Split with and without 2

1 2 5 9 13 1 2 3 4 13 1 2 7 19 1 3 5 9 1 4 7 13 19

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1

Introduction

2

First test

3

Final program

4

Results

5

Conclusion

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Programs

Independent set method Independent set method (Gunnar Brinkmann) General clique solver (Patric ¨ Osterg˚ ard)

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Determine correctness

Compare some specific graphs Compare the number of graphs per diameter, given an upper bound for uredop

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Best graphs

Diameter after Restrictions Diameter (D) edge removal (E) uredop value D 4 4 5 0.92423 D 4, E 4 4 4 0.93317 D 3 3 5 0.96598 D 3, E 4 3 4 0.97854

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1

Introduction

2

First test

3

Final program

4

Results

5

Conclusion

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For our practical problem

Best graphs within the hour Much better results than those found by Lockheed Martin

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Time needed depends heavily on the input weights.

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Possible variations of input:

Different amount of vertices Different values weight function Different amount of connections per vertex Different GPos

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Thank you!

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